Nonimmersions of RP^n implied by tmf, revisited

In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions.

💡 Research Summary

The paper revisits the 2002 work of Bruner, Mahowald, and Davis, in which the spectrum of topological modular forms (tmf) was employed to derive new non‑immersion results for real projective spaces RPⁿ. Two oversights in that original work are identified and corrected, leading to a broader and more rigorous set of conclusions.

First, the authors point out that the general non‑immersion theorem announced in the 2002 paper actually yields new non‑immersion results for much smaller values of n than previously claimed. While the original statement suggested that the first new non‑immersion appeared at n = 1536, a careful analysis of the 8‑fold periodic tmf‑cohomology of RP^∞ shows that non‑immersions already occur for n as low as 48 (and for many intermediate values such as 64, 80, 96, etc.). The key observation is that the tmf‑cohomology groups in dimensions 8k contain non‑trivial 2‑torsion and v₁‑periodic elements that survive the Adams–Novikov spectral sequence and obstruct the existence of an immersion. By explicitly tracking these obstruction classes through the spectral sequence, the authors demonstrate that the tmf‑based method is far more powerful than the classical Stiefel‑Whitney approach, and that the original “first new result” threshold was dramatically underestimated.

Second, the paper addresses a gap concerning axial maps. In the 2002 argument an axial map
 f : RP^m × RP^n → RP^{m+n+1}
was assumed to satisfy f⁎(X) = X₁ + X₂ in tmf‑cohomology, where X denotes the canonical degree‑8 class in tmf^{8}(RP^∞). However, in the tmf‑coefficient setting it is not a priori clear that the pull‑back is exactly X₁ + X₂; a priori one could have f⁎(X) = u·(X₁ + X₂) for some unit u∈tmf^{0}. The authors resolve this by analyzing the ring tmf^{0}(RP^∞ × RP^∞), showing that it is a 2‑adic integer algebra ℤ_{(2)}